Scott Mutchnik : A new $\mathrm{NSOP}_2$-$\mathrm{SOP}_3$ dichotomy

Posted by Scott Mutchnik , part of the Logic Seminar.

At

April 21, 2026, 3 p.m.

In

636 SEO

Abstract

Following the resolution of the longstanding formerly open question of whether the 1-strict order property, $\mathrm{SOP}_1$, is equal to $\mathrm{SOP}_2$, one of the main problems in model theory is to determine whether $\mathrm{SOP}_2$ is equal to $\mathrm{SOP}_3$. In this talk we discuss our very recent progress on this question, where we show, for $\mathcal{H}$ a hereditary class defined by finitely many forbidden weakly embedded substructures, that if the theory of every structure of which $\mathcal{H}$ is the age has $\mathrm{SOP}_2$, then the theory of every structure of which $\mathcal{H}$ is the age has $\mathrm{SOP}_3$. Crucially, this is a strict dichotomy: it is false if we replace $\mathrm{SOP}_2$ with the tree property (as demonstrated in examples of Conant and Kruckman, and Kruckman and Ramsey). We will start with some historical background on the problem of whether $\mathrm{NSOP}_2$ is equal to $\mathrm{NSOP}_3$, as well as the context for our theorem within the setting of $\mathrm{NSOP}_r$ theories for $r$ a real number. We will see how, by modifying recent work of Bodirsky, Bodor and Marimon, our general results reduce to what initially superficially appeared to be a mere verification for a concrete family of examples (the generic structures of Cherlin, Shelah and Shi). If time permits, we will give a very rough overview of the proof of this theorem, which involves harder versions of arguments on the real-valued $\mathrm{NSOP}_r$ hierarchy.